I. Introduction

II. Methods

Notation:

\[ \begin{aligned} k:& \text{subject}, k = 1,...,180\\ j:& \text{sequence}, j=0,1,2\\ i:& \text{period}, i=1,2,3\\ h:& \text{week}, h=1,2,3,4\\ \pi:& \text{period effect}\\ \tau:& \text{treatment effect}\\ \alpha:& \text{grouped treatment effects}\\ \beta:& \text{demographic effects}\\ \lambda:& \text{sequence/carryover effect}\\ \gamma:& \text{week effect}\\ b:& \text{subject-specific intercepts}\\ \mu:& \text{fixed-effect intercept}\\ \varepsilon:& \text{residual error}\\ \end{aligned} \]

III. Results

3.1 Primary Objective 1

Proportion of Adverse Events per period

Patient Distribution: 30 people in per treatment sequence Initial Observation: Most patients do not face adverse events over the course of 4 weeks in each of the consecutive periods.

add demographic data

demographic summary

Distributional Summary of Predictor variables in the Study
period1
period2
period3
Characteristic No
N = 164 		Yes
N = 16 		p-value No
N = 168 		Yes
N = 12 		p-value No
N = 163 		Yes
N = 17 		p-value
age 31 (25, 37) 42 (40, 44) <0.001 32 (25, 38) 40 (37, 44) <0.001 32 (25, 38) 38 (32, 42) 0.012
1 Median (Q1, Q3)
2 Wilcoxon rank sum test

GLMM

Model Selection

LRT Between Model w/ and w/o Interaction Between Tx and Period
term npar AIC BIC logLik minus2logL statistic df p.value
glmm2 12 278.67 330.17 -127.34 254.67 NA NA NA
glmm1 16 281.34 350.01 -124.67 249.34 5.33 4 0.25
LRT Between Model w/ and w/o Race, Gender and Carryover Effect
term npar AIC BIC logLik minus2logL statistic df p.value
glmm3 7 271.25 301.29 -128.62 257.25 NA NA NA
glmm2 12 278.67 330.17 -127.34 254.67 2.57 5 0.77

Model Result (with Carryover Effect)

Characteristic OR 95% CI p-value
(Intercept) 0.00 0.00, 0.00 <0.001
Treatment


    Pill A — —
    Gel B 1.20 0.55, 2.64 0.6
    Gel C 0.84 0.37, 1.92 0.7
Period


    period1 — —
    period2 0.71 0.31, 1.64 0.4
    period3 1.13 0.52, 2.46 0.8
Treatment Sequence


    ABC or BAC — —
    CAB or ACB 0.61 0.23, 1.62 0.3
    BCA or CBA 0.84 0.35, 2.02 0.7
age 1.19 1.11, 1.27 <0.001
gender


    Female — —
    Male 0.90 0.43, 1.89 0.8
race


    black — —
    others 1.55 0.61, 3.95 0.4
    white 1.74 0.65, 4.68 0.3
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

Final Model Result (without Carryover Effect)

Notation: \[ \begin{aligned} \widehat{logit}(P(AE_{ijk} = 1)) &= b_{k} + \mu +\tau_{i} +\pi_j + \beta{a_{k}},\\ &\quad i = 1, 2, 3,\ j = 1, 2, 3,\ k = (1, 2, \ldots, 180),\\ &\quad b_{k} \sim N(0, {{\sigma}^2}_b) \end{aligned} \]

Characteristic OR 95% CI p-value
(Intercept) 0.00 0.00, 0.00 <0.001
Treatment


    Pill A — —
    Gel B 1.18 0.54, 2.58 0.7
    Gel C 0.85 0.37, 1.94 0.7
Period


    period1 — —
    period2 0.72 0.31, 1.65 0.4
    period3 1.11 0.51, 2.40 0.8
age 1.19 1.12, 1.27 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

AE Prediction

Model Diagnostics

3.2 Primary Objective 2

Model

$$

\[\begin{align} & Y_{hik} \sim \text{Binomial}\big(n=7, p_{hik}\big), \\[1em] & \text{logit}(p_{hik}) = \mu + b_k + \pi_i + \tau_i + \lambda_{j} + \gamma*h, \\[0.5em] & b_k \sim \mathcal{N}(0, \sigma_b^2) \end{align}\]

$$

GLMM Results of Treatment on Adherence
Characteristic OR 95% CI p-value
Treatment


    Pill A — —
    Gel B 0.14 0.12, 0.17 <0.001
    Gel C 0.21 0.17, 0.24 <0.001
Week 0.73 0.69, 0.76 <0.001
Period


    1 — —
    2 1.15 1.00, 1.32 0.051
    3 1.05 0.92, 1.20 0.5
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

3.3 Secondary Objective 1

Table 1. Baseline Characteristics
Characteristic Overall
N = 1801		 ABC
N = 301		 ACB
N = 301		 BAC
N = 301		 BCA
N = 301		 CAB
N = 301		 CBA
N = 301		 p-value2
Age (years) 32 (8) [18, 45] 33 (8) [19, 44] 31 (7) [18, 42] 31 (7) [20, 44] 33 (8) [19, 45] 32 (7) [21, 44] 31 (7) [20, 44] 0.6
Race






0.9
    Black 64 (36%) 12 (40%) 13 (43%) 12 (40%) 10 (33%) 8 (27%) 9 (30%)
    White 52 (29%) 8 (27%) 9 (30%) 7 (23%) 7 (23%) 12 (40%) 9 (30%)
    Other 64 (36%) 10 (33%) 8 (27%) 11 (37%) 13 (43%) 10 (33%) 12 (40%)
Female 81 (45%) 13 (43%) 11 (37%) 16 (53%) 14 (47%) 12 (40%) 15 (50%) 0.8
1 Mean (SD) [Min, Max]; n (%)
2 Kruskal-Wallis rank sum test; Pearson’s Chi-squared test

preliminary assessment and comparison of systemic and local Pharmacokinetics (PK) of Pill A, Gel B, and Gel C and the correlation of PK with adherence measures and the occurrence of adverse events

Blood PK

\[ Y_{ik} = \mu + b_{k} + \pi_i + \tau_{i} + \lambda_{i} + \alpha_{ik} + \beta_{ik} + \varepsilon_{ik}, \quad b_{k} \sim N(0, \sigma_b^2), \quad \varepsilon_{ik} \sim N(0, \sigma^2) \]

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: bloodVL_change ~ treatment + overall_adhere + overall_safety +  
##     period + (1 | ptid)
##    Data: merged
## 
## REML criterion at convergence: 4827.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.5936 -0.5795  0.0153  0.6127  3.0873 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  ptid     (Intercept)  23.09    4.805  
##  Residual             445.60   21.109  
## Number of obs: 540, groups:  ptid, 180
## 
## Fixed effects:
##                                Estimate Std. Error       df t value
## (Intercept)                     73.4552     6.7058 419.7422  10.954
## treatmentGel B                  -6.7916     2.3555 417.1354  -2.883
## treatmentGel C                  -9.0880     2.2967 391.5595  -3.957
## overall_adhere                   0.9365     0.1882 269.4894   4.975
## overall_safetyNo Adverse Event  26.4624     3.4675 521.9424   7.632
## periodperiod2                   -2.3295     2.2271 356.0129  -1.046
## periodperiod3                   -1.5433     2.2252 355.4545  -0.694
##                                            Pr(>|t|)    
## (Intercept)                    < 0.0000000000000002 ***
## treatmentGel B                              0.00414 ** 
## treatmentGel C                    0.000090106833765 ***
## overall_adhere                    0.000001161748202 ***
## overall_safetyNo Adverse Event    0.000000000000111 ***
## periodperiod2                               0.29629    
## periodperiod3                               0.48842    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) trtmGB trtmGC ovrll_ ov_NAE prdpr2
## treatmntGlB -0.437                                   
## treatmntGlC -0.359  0.538                            
## overall_dhr -0.832  0.328  0.247                     
## ovrll_sfNAE -0.616  0.080  0.032  0.196              
## periodperd2 -0.129 -0.009 -0.006 -0.026 -0.038       
## periodperd3 -0.173  0.001  0.001  0.004  0.009  0.499
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 73.5 60.3, 86.6 <0.001
Treatment


    Gel B vs Pill A -6.8 -11.4, -2.2 0.004
    Gel C vs Pill A -9.1 -13.6, -4.6 <0.001
Additional Day of Adherence 0.9 0.6, 1.3 <0.001
Adverse Event


    No Adverse Event 26.5 19.7, 33.3 <0.001
Period


    Period 2 vs Period 1 -2.3 -6.7, 2.1 0.3
    Period 3 vs Period 1 -1.5 -5.9, 2.8 0.5
Abbreviation: CI = Confidence Interval 
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     6.79 2.36 417   2.882  0.0115
##  Pill A - Gel C     9.09 2.30 391   3.956  0.0003
##  Gel B - Gel C      2.30 2.24 359   1.027  0.5602
## 
## Results are averaged over the levels of: overall_safety, period 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2    2.330 2.23 355   1.046  0.5484
##  period1 - period3    1.543 2.23 355   0.694  0.7674
##  period2 - period3   -0.786 2.23 356  -0.353  0.9337
## 
## Results are averaged over the levels of: treatment, overall_safety 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates

Skin PK

Linear Model

## 
## Call:
## lm(formula = skinVL_change ~ treatment + overall_adhere + overall_safety + 
##     period, data = merged)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -112.388  -22.128    1.035   20.921  115.662 
## 
## Coefficients:
##                                Estimate Std. Error t value             Pr(>|t|)
## (Intercept)                    105.6003     9.9680  10.594 < 0.0000000000000002
## treatmentGel B                 -24.4131     3.6314  -6.723       0.000000000046
## treatmentGel C                 -36.9951     3.5493 -10.423 < 0.0000000000000002
## overall_adhere                   1.8154     0.2753   6.594       0.000000000103
## overall_safetyNo Adverse Event  55.7940     5.2166  10.696 < 0.0000000000000002
## periodperiod2                    0.8567     3.4535   0.248                0.804
## periodperiod3                    1.6225     3.4507   0.470                0.638
##                                   
## (Intercept)                    ***
## treatmentGel B                 ***
## treatmentGel C                 ***
## overall_adhere                 ***
## overall_safetyNo Adverse Event ***
## periodperiod2                     
## periodperiod3                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 32.74 on 533 degrees of freedom
## Multiple R-squared:  0.3581, Adjusted R-squared:  0.3508 
## F-statistic: 49.55 on 6 and 533 DF,  p-value: < 0.00000000000000022
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 105.6 86.0, 125.2 <0.001
Treatment


    Gel B vs Pill A -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 1.8 1.3, 2.4 <0.001
Adverse Event


    No Adverse Event 55.8 45.5, 66.0 <0.001
Period


    Period 2 vs Period 1 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval 
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     24.4 3.63 533   6.723 <0.0001
##  Pill A - Gel C     37.0 3.55 533  10.423 <0.0001
##  Gel B - Gel C      12.6 3.47 533   3.631  0.0009
## 
## Results are averaged over the levels of: overall_safety, period 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2   -0.857 3.45 533  -0.248  0.9666
##  period1 - period3   -1.623 3.45 533  -0.470  0.8853
##  period2 - period3   -0.766 3.45 533  -0.222  0.9733
## 
## Results are averaged over the levels of: treatment, overall_safety 
## P value adjustment: tukey method for comparing a family of 3 estimates

Linear Mixed Model

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: skinVL_change ~ treatment + overall_adhere + overall_safety +  
##     period + (1 | ptid)
##    Data: merged
## 
## REML criterion at convergence: 5269.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.4333 -0.6760  0.0316  0.6391  3.5333 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  ptid     (Intercept)    0      0.00   
##  Residual             1072     32.74   
## Number of obs: 540, groups:  ptid, 180
## 
## Fixed effects:
##                                Estimate Std. Error       df t value
## (Intercept)                    105.6003     9.9680 533.0000  10.594
## treatmentGel B                 -24.4131     3.6314 533.0000  -6.723
## treatmentGel C                 -36.9951     3.5493 533.0000 -10.423
## overall_adhere                   1.8154     0.2753 533.0000   6.594
## overall_safetyNo Adverse Event  55.7940     5.2166 533.0000  10.696
## periodperiod2                    0.8567     3.4535 533.0000   0.248
## periodperiod3                    1.6225     3.4507 533.0000   0.470
##                                            Pr(>|t|)    
## (Intercept)                    < 0.0000000000000002 ***
## treatmentGel B                       0.000000000046 ***
## treatmentGel C                 < 0.0000000000000002 ***
## overall_adhere                       0.000000000103 ***
## overall_safetyNo Adverse Event < 0.0000000000000002 ***
## periodperiod2                                 0.804    
## periodperiod3                                 0.638    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) trtmGB trtmGC ovrll_ ov_NAE prdpr2
## treatmntGlB -0.429                                   
## treatmntGlC -0.354  0.534                            
## overall_dhr -0.826  0.311  0.234                     
## ovrll_sfNAE -0.627  0.079  0.033  0.206              
## periodperd2 -0.137 -0.008 -0.005 -0.025 -0.037       
## periodperd3 -0.180  0.001  0.001  0.004  0.009  0.499
## optimizer (nloptwrap) convergence code: 0 (OK)
## boundary (singular) fit: see help('isSingular')
Comparison Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 105.6 86.0, 125.2 <0.001
Treatment


    Gel B vs Pill A -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 1.8 1.3, 2.4 <0.001
Adverse Event


    No Adverse Event 55.8 45.5, 66.0 <0.001
Period


    Period 2 vs Period 1 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval 
##  contrast       estimate   SE  df t.ratio p.value
##  Pill A - Gel B     24.4 3.63 412   6.721 <0.0001
##  Pill A - Gel C     37.0 3.55 388  10.422 <0.0001
##  Gel B - Gel C      12.6 3.47 359   3.631  0.0009
## 
## Results are averaged over the levels of: overall_safety, period 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates
##  contrast          estimate   SE  df t.ratio p.value
##  period1 - period2   -0.857 3.45 356  -0.248  0.9666
##  period1 - period3   -1.623 3.45 355  -0.470  0.8853
##  period2 - period3   -0.766 3.45 356  -0.222  0.9733
## 
## Results are averaged over the levels of: treatment, overall_safety 
## Degrees-of-freedom method: kenward-roger 
## P value adjustment: tukey method for comparing a family of 3 estimates

Combined Table

Comparison
Blood Viral Load
Skin Viral Load
Beta 95% CI p-value Beta 95% CI p-value
Intercept (Pill A, Period 1, Sequence 0, No Adherence or AE) 73.5 60.3, 86.6 <0.001 105.6 86.0, 125.2 <0.001
Treatment





    Gel B vs Pill A -6.8 -11.4, -2.2 0.004 -24.4 -31.5, -17.3 <0.001
    Gel C vs Pill A -9.1 -13.6, -4.6 <0.001 -37.0 -44.0, -30.0 <0.001
Additional Day of Adherence 0.9 0.6, 1.3 <0.001 1.8 1.3, 2.4 <0.001
Adverse Event





    No Adverse Event 26.5 19.7, 33.3 <0.001 55.8 45.5, 66.0 <0.001
Period





    Period 2 vs Period 1 -2.3 -6.7, 2.1 0.3 0.9 -5.9, 7.6 0.8
    Period 3 vs Period 1 -1.5 -5.9, 2.8 0.5 1.6 -5.2, 8.4 0.6
Abbreviation: CI = Confidence Interval

3.4 Secondary Objective 2

Model

$$

\[\begin{align} & Y_{hik} \sim \text{Binomial}\big(n=7, p_{hik}\big), \\[1em] & \text{logit}(p_{hik}) = \mu + b_k + \pi_i + \lambda_{j} + \gamma*h + \beta_d *d_k + \beta_r * regimen + \beta_p *product\\ & + \beta_{dr} *d_k * regimen + \beta_{dp} *d_k * product \\[0.5em] & b_k \sim \mathcal{N}(0, \sigma_b^2) \end{align}\]

$$

Interaction Results of Demographic Variables on Adherence
Characteristic OR 95% CI p-value
Age 1.21 1.18, 1.24 <0.001
Product


    Gel — —
    Pill 4.35 2.98, 6.35 <0.001
Gender


    Male — —
    Female 1.36 0.95, 1.95 0.10
Race


    white — —
    black 1.02 0.65, 1.60 >0.9
    others 0.97 0.62, 1.53 0.9
Regimen


    once — —
    three 0.76 0.58, 1.00 0.050
Period


    1 — —
    2 1.14 0.99, 1.32 0.073
    3 1.03 0.89, 1.18 0.7
Week 0.73 0.69, 0.76 <0.001
Age * Product


    Age * Pill 1.00 0.97, 1.03 >0.9
Product * Gender


    Pill * Female 1.00 0.71, 1.40 >0.9
Product * Race


    Pill * black 1.23 0.81, 1.86 0.3
    Pill * others 1.05 0.69, 1.61 0.8
Age * Regimen


    Age * three 1.02 1.00, 1.04 0.074
Gender * Regimen


    Female * three 0.89 0.69, 1.15 0.4
Race * Regimen


    black * three 1.14 0.84, 1.56 0.4
    others * three 1.02 0.74, 1.41 0.9
Abbreviations: CI = Confidence Interval, OR = Odds Ratio
Main Effect Results of Demographic Variables on Adherence
Characteristic OR 95% CI p-value
Age 1.22 1.19, 1.25 <0.001
Gender


    Male — —
    Female 1.29 0.92, 1.80 0.14
Race


    white — —
    black 1.11 0.73, 1.69 0.6
    others 0.99 0.65, 1.50 >0.9
Regimen


    once — —
    three 0.71 0.63, 0.80 <0.001
Product


    Gel — —
    Pill 4.84 4.12, 5.70 <0.001
Period


    1 — —
    2 1.14 0.99, 1.32 0.065
    3 1.04 0.91, 1.20 0.6
Week 0.73 0.69, 0.76 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio

IV. Discussion

V. Conclusion

VI. Appendix